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<p><dfn class="terminology">Proof</dfn> (i) Necessary ConditionSuppose that (<a href="" class="xref" data-knowl="./knowl/eq2_25.html" title="Equation 2.6.1">(2.6.1)</a>) is an exact ODE, i. e., (<a href="" class="xref" data-knowl="./knowl/eq2_26.html" title="Equation 2.6.2">(2.6.2)</a>) is satisfied, we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_25.html ./knowl/eq2_26.html ./knowl/eq2_27.html">
\begin{equation*}
M(x, y)=\frac{\partial \Psi}{\partial x},\quad N(x, y)=\frac{\partial \Psi}{\partial y}~\rightarrow~
\frac{\partial M}{\partial y}=\frac{\partial^2 \Psi}{\partial y \partial x},~\quad~
\frac{\partial N}{\partial x}=\frac{\partial^2 \Psi}{\partial y \partial x},~\rightarrow~
\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}.
\end{equation*}
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<p class="continuation">So (<a href="" class="xref" data-knowl="./knowl/eq2_27.html" title="Equation 2.6.3">(2.6.3)</a>) is satisfied.</p>
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